The following is an addition to A function from partitions to natural numbers - is it familiar? the function $f(\lambda)$ as defined there, when summed over all partitions of n, gives an unexpected hit in http://oeis.org/A232434. Checked up to n=36. In short:
define g(x, q) by
d/dx g(x, q) == g(x, q)^2 * g(q* x, q),
with series g(x,q)==sum( k=0..$\infty$; $c_{k}$(q) x^k/k! ),
define $t_{n}$=n(n+1)/2; then the coefficient of q^$t_{n-1}$ in $c_{n+1}(q)$ is
sum( { | $(\lambda)$ | = n }; $f(\lambda)$ ) ;

examples:
f(n=2)=6 and $c_{3}$(q) =6 + (6 q) + 2 q^2 + q^3 ; q^$t_{1}$ = q^1 ; coeff. =6;
f(n=3)=14 and $c_{4}$(q) =24 + 36 q + 22 q^2 + (14 q^3) + 6 q^4 + 2 q^5 + q^6 ; q^$t_{2}$ = q^3 ; coeff. =14;

Remark that the differential equation has no closed form solution, but its series can be generated upto x^n for any given n.

Question: are there any arguments to trust\distrust this conjecture? Can it be proved ?